Is it possible to only see contacts with phone numbers?

The native Contacts and Phone apps do not support this in iOS. As a workaround, you could use a third party contacts manager which supports smart groups, creating “Has a phone number” and “Has an email address” groups and working from these. Note that the contact lookup sheets shown by other apps (like, say, Mail) are provided by the native app and thus will not show your new groups.

A free example of such an app for the iPhone would be Tacts – searching for “smart groups” on the App Store should get you more.

The iPhone Contacts app will not let you do this. You have a few options though. One, just add the people you call a lot to your Favorites list in Contacts. The other is to rely more on searching for a contact (by scrolling to the very top of the screen) rather than swiping. I do this almost exclusively versus scrolling through the list, as I've found it much faster.

You could use Gmail contacts for your contacts you need to email, and your iCloud for the people you need to phone. Then, switch off Gmail syncing on your iphone, and switch off iCloud on your iPad. If you ever need to email someone you can switch on GMail contacts on your iPhone, send the email, then switch it off again.

It's a little clumsy because you end up with duplicates, but that's not really a problem as such, as you can always consolidate them into one address book in future.

Arrange your contacts in Gmail

There will be (My Contacts) with all numbers and emails

Choose only contacts with emails and put them in a new group (say MyGmail Contacts)

Now you will have only contacts with numbers in the main (My Contacts) folder

Syc with iPhone

Now you will have only contacts with numbers in iPhone